Intermediate Techniques: Boxes and Pairs

When no single is left, remove impossible notes first. Hard puzzles use these four techniques.

What these techniques do

Once the singles run out, there are usually many empty cells left, each with two, three or more possibilities. Now you need a different mindset: instead of placing a number, prove that some notes are impossible and remove them. Remove enough and a cell is left with one number, or a number with one place, and you are back to singles.

Every diagram on this page shows notes: green cells are the key to the technique, bold numbers are the key numbers, and notes with a red slash can be removed.

Pointing pair

In some box, all the notes for a number d lie in the same row (or column). Wherever d ends up inside those cells, that row will use up its d, so no other cell of that row outside the box can be d.

Box/line reduction

The reverse: in some row (or column), all the notes for d lie inside one box. Then the box's d must be in that row, and the d notes in the box's other rows can be removed.

These two are two sides of the same idea: both look at the three cells where a box and a row or column overlap.

Naked pair

In a row, column or box, two cells have exactly the same two notes, say only 1 and 9. One of them must be 1 and the other 9; you just do not know which yet. So no other cell in that area can be 1 or 9.

Hidden pair

In one area, two numbers can only go in the same two cells. Those two cells are taken by those two numbers, so every other note in them can be removed.

Hidden pairs are harder to see than naked pairs, because the two cells also contain other notes. Look for them from the numbers: ask one by one, "Where else can this number go in this box?"

Practice

These four techniques plus singles solve every Hard puzzle. When you get stuck, press Hint: it names the technique and marks the cells involved, which is easiest to follow alongside the diagrams on this page.

Next: Advanced techniques: triples, X-Wing and the wings.